Classical Mechanics · Advanced Topics

Plasma Physics

Plasma — the fourth state of matter — is an ionized gas of electrons and ions. It makes up 99% of the visible universe: stars, nebulae, the solar wind, and lightning. Understanding plasma is key to fusion energy and space weather.

PrerequisitesElectromagnetism(Ch.14–15)⋅Maxwell′sequations(Ch.EM)⋅Statisticalmechanics(Ch.SElectromagnetism (Ch. 14–15) \cdot Maxwell's equations (Ch. EM) \cdot Statistical mechanics (Ch. S) ⋅Fluidmechanics(Ch.FM\cdot Fluid mechanics (Ch. FM
Learning Goals
  • State the three conditions that define a plasma (Debye shielding, quasi-neutrality, collective behaviour) and compute the Debye length for given n and T.
  • Describe Larmor (cyclotron) motion and calculate the cyclotron frequency and Larmor radius for electrons and protons in a given magnetic field.
  • DerivethedispersionrelationforEMwavesinaplasmaandexplainwhyfrequenciesbelowωerive the dispersion relation for EM waves in a plasma and explain why frequencies below \omegap are reflected.
  • Write the ideal MHD equations and explain flux-freezing (Alfvén's theorem) and Alfvén wave propagation.
  • StatetheLawsoncriterionandexplainthetripleproductnTτEasthepracticalfigureofState the Lawson criterion and explain the triple product nT\tau_E as the practical figure of merit for fusion ignition.

PL.1 What is a Plasma?

Definition PL.1 — Plasma Conditions
An ionized gas becomes a true plasma when three conditions are met:1. Collective behavior dominates: TheDebyelengthλD=(ε0kBT/(ne2))satisfiesλD≪L(systemsize).WithinλD,electrThe Debye length \lambda_D = \sqrt(\varepsilon_{0}k_{BT}/(ne^{2})) satisfies \lambda_D ≪ L (system size). Within \lambda_D, electrstatic shielding occurs.2. Many particles per Debye sphere:nλD3≫1(quasi−neutralityholdsonscales>λDn \lambda_D^{3} ≫ 1 (quasi-neutrality holds on scales > \lambda_D.3. Plasma frequency dominates collisions: ωpτ≫1,whereωp=(ne2/(ε0me))andτisthecollisiontime\omega_p \tau ≫ 1, where \omega_p = \sqrt(ne^{2}/(\varepsilon_{0}m_{e})) and \tau is the collision time

The Debye length is the key scale:

λD=ε0kBTne2(Debye length)\lambda_D=\sqrt{\frac{\varepsilon_0 k_BT}{ne^2}} \qquad \text{(Debye length)}(PL.1)

The potential of a test charge Q in a plasma: φ = (Q/4πε₀r) × e^(−r/λ_D). Beyond λ_D, the plasma screens the charge completely. For solar wind (T ≈ 10⁵ K, n ≈ 10⁷ m⁻³): λ_D ≈ 7 m. For fusion plasma (T ≈ 10⁸ K, n ≈ 10²⁰ m⁻³): λ_D ≈ 70 μm.

PL.2 Single-Particle Motion

A charged particle (charge q, mass m) in crossed E and B fields:

mdvdt=q(E+v×B)(Lorentz force)m\frac{d\mathbf{v}}{dt}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}) \qquad \text{(Lorentz force)}(PL.2)

In a pure magnetic field B = Bẑ, the particle undergoes Larmor (cyclotron) motion— circular orbit in the plane perpendicular to B with:

ωc=qBm(cyclotron frequency)rL=mv⊥∣q∣B(Larmor radius)\omega_c=\frac{qB}{m} \qquad \text{(cyclotron frequency)} \qquad r_L=\frac{mv_\perp}{|q|B} \qquad \text{(Larmor radius)}(PL.3)

For electrons: ω_ce = eB/m_e ≈ 1.76×10¹¹ B rad/s. For protons: ω_ci = eB/m_p ≈ 9.58×10⁷ B rad/s (1836× smaller). In the Earth's field (B ≈ 5×10⁻⁵ T): electron cyclotron frequency ≈ 1.4 MHz (radio), proton ≈ 760 Hz (ELF).

With crossed E ⊥ B fields, the guiding center drifts perpendicular to both:

vE=E×BB2(E×B drift — same for all charges)\mathbf{v}_E=\frac{\mathbf{E}\times\mathbf{B}}{B^2} \qquad \text{(}\mathbf{E}\times\mathbf{B}\text{ drift — same for all charges)}(PL.4)

Because E×B drift is charge-independent, electrons and ions drift together — no net current. Other drifts (gradient-B, curvature) are charge-dependent and drive currents.

Example PL.1 — Cyclotron Motion in Earth's Field

Anelectronwithkineticenergy1keVenterstheEarth′smagneticequator(B=3×10−5TAn electron with kinetic energy 1 keV enters the Earth's magnetic equator (B = 3\times10^{-5} T. Find the Larmor radius and cyclotron frequency.

Velocity:KE=12mev2→v=(2×1000×1.6×10−19/9.11×10−31)=(3.51×1014)=1.87×107m/s(non−relatKE = \frac{1}{2}m_{e} v^{2} \to v = \sqrt(2\times1000\times1.6\times10^{-19}/9.11\times10^{-31}) = \sqrt(3.51\times10^{14}) = 1.87\times10^{7} m/s (non-relativistic:v/c=6.3ivistic: v/c = 6.3%.
Larmor radius:rL=mev/(eB)=9.11×10−31×1.87×107/(1.6×10−19×3×10−5)=1.70×10−23/4.8×10−24≈3r_{L} = m_{e} v/(eB) = 9.11\times10^{-31} \times 1.87\times10^{7} / (1.6\times10^{-19} \times 3\times10^{-5}) = 1.70\times10^{-23}/4.8\times10^{-24} \approx 3.6 m.
Cyclotron frequency:ωc=eB/me=1.6×10−19×3×10−5/9.11×10−31=5.27×106rad/s→f=838kHz(AMradioband\omega_c = eB/m_{e} = 1.6\times10^{-19} \times 3\times10^{-5}/9.11\times10^{-31} = 5.27\times10^{6} rad/s \to f = 838 kHz (AM radio band).
Physical context:Energetic electrons spiral along field lines in the Van Allen belts, bouncing between mirror points near the poles. Their cyclotron radiation (whistler waves) propagates along field lines.

PL.3 Plasma Waves

Plasmas support a rich variety of waves. The simplest: plasma (Langmuir) oscillations. Displace all electrons by δx while ions are fixed: restoring force from charge separation → oscillations at the plasma frequency:

ωp=ne2ε0me(plasma frequency)\omega_p=\sqrt{\frac{ne^2}{\varepsilon_0m_e}} \qquad \text{(plasma frequency)}(PL.5)

EM waves in a plasma have the dispersion relation:

ω2=ωp2+c2k2(electromagnetic waves in plasma)\omega^2=\omega_p^2+c^2k^2 \qquad \text{(electromagnetic waves in plasma)}(PL.6)

For ω < ω_p: k is imaginary — the wave is evanescent (reflected). This explains why AM radio waves bounce off the ionosphere (ω_p ∼ 10–30 MHz for the F layer). For ω > ω_p: wave propagates, with phase velocity v_ph = ω/k > c and group velocity v_g = dω/dk = c²k/ω < c (information travels at v_g).

The index of refraction for a plasma: n = ck/ω = √(1 − ω_p²/ω²). At ω_p: n → 0 (total reflection). This is used in magnetic confinement: microwaves probe the plasma density because their cutoff frequency equals ω_p.

PL.4 Magnetohydrodynamics (MHD)

When the plasma behavior is collective (many particles), we describe it as a conducting fluid — magnetohydrodynamics. The key equations:

∂ρ∂t+∇⋅(ρv)=0(continuity)\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0 \qquad \text{(continuity)}(PL.7)
ρ(∂v∂t+v⋅∇v)=J×B−∇P(MHD momentum)\rho\left(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right)=\mathbf{J}\times\mathbf{B}-\nabla P \qquad \text{(MHD momentum)}(PL.8)
∂B∂t=∇×(v×B)−1μ0σ∇2B(induction equation)\frac{\partial\mathbf{B}}{\partial t}=\nabla\times(\mathbf{v}\times\mathbf{B})-\frac{1}{\mu_0\sigma}\nabla^2\mathbf{B} \qquad \text{(induction equation)}(PL.9)

The induction equation describes flux freezing: in ideal MHD (σ → ∞), ∂B/∂t = ∇×(v×B) — magnetic field lines are frozen into the conducting fluid and move with it. This is Alfvén's theorem (Nobel 1970). The magnetic Reynolds number Rm = μ₀σvL governs whether diffusion (Rm ≪ 1) or advection (Rm ≫ 1) dominates.

Alfvén waves: transverse perturbations propagating along B at:

vA=Bμ0ρ(Alfveˊn speed)v_A=\frac{B}{\sqrt{\mu_0\rho}} \qquad \text{(Alfvén speed)}(PL.10)

In the solar wind (B ≈ 5 nT, ρ ≈ 10⁻²⁰ kg/m³): v_A ≈ 40 km/s. In the solar corona (B ≈ 100 G, n ≈ 10¹⁴ m⁻³): v_A ≈ 10⁴ km/s ≈ 3% c.

PL.5 Fusion Plasmas

The Lawson criterion for D-T fusion (nτE ≥ 10²⁰ m⁻³·s at T ≈ 10⁸ K) requires simultaneously high density n, confinement time τ_E, and temperature T. The triple productnTτ_E > 3×10²¹ keV·m⁻³·s is the practical figure of merit.

Tokamak geometry: toroidal solenoid with poloidal field from plasma current → helical field lines. Plasma pressure balance: β = nk_BT/(B²/2μ₀) ≈ 5–10%. Confinement: τ_E ∝ B^1.8 R^1.97 (Bohm/gyro-Bohm scaling — still not fully understood). ITER (under construction): designed to achieve Q = P_fusion/P_input ≥ 10 (first device).

Definition PL.2 — Common Traps
  • Ionized gas is not automatically plasma: collective behavior and Debye shielding must dominate.
  • Quasi-neutral does not mean charge-free: small charge separations drive plasma oscillations and waves.
  • Cyclotron sign matters: electrons and ions gyrate in opposite senses.
  • E\timesBdriftischargeindependentE\timesB drift is charge independent: both signs drift together, so it does not by itself create current.
  • MHD averages over particles: it fails when kinetic effects, collisions, or small scales dominate.
Exercises — PL.1–PL.5 Plasma Physics
1.
CalculatetheDebyelengthλD(\mum)forafusionplasmaatT=108K,n=1020m−3Calculate the Debye length \lambda_D (\mum) for a fusion plasma at T = 10^{8} K, n = 10^{20} m^{-3}.
μm
Straightforward
2.
ExplainwhyAMradiobouncesofftheionospherebutFMdoesnot.Whatelectrondensity(m−Explain why AM radio bounces off the ionosphere but FM does not. What electron density (m^{-}³) is needed for a 10 MHz cutoff?
m⁻³
Intermediate
3.Describethemagneticmirroreffect.WhatisthelossconeangleforamirrorratioR=10Describe the magnetic mirror effect. What is the loss cone angle for a mirror ratio R = 10? How do the Van Allen belts act as a magnetic mirror?
Intermediate
4.Estimate theAlfveˊnspeedandtoroidalAlfveˊneigenmode(TAE)frequencyinanITER−liketokamak(R=he Alfvén speed and toroidal Alfvén eigenmode (TAE) frequency in an ITER-like tokamak (R =5T,deuteriumplasman=1020m−3).WhydoTAEsmatterforfusion5 T, deuterium plasma n = 10^{20} m^{-3}). Why do TAEs matter for fusion?
Challenging
Key Takeaways
  • Plasmaconditions:λD≪L,ND≫1,ωpτ≫1.DebyeshieldingscreenschargeoverλDPlasma conditions: \lambda_D ≪ L, N_{D} ≫ 1, \omega_p \tau ≫ 1. Debye shielding screens charge over \lambda_D.
  • Cyclotronmotion:ωc=qB/m,rL=mv⊥/(qB).E\timesBdriftischarge−independentCyclotron motion: \omega_c = qB/m, r_{L} = mv⊥/(qB). E\timesB drift is charge-independent.
  • EMwaves:ω2=ωp2+c2k2.ReflectionbelowωpexplainsionosphericradiobounceEM waves: \omega^{2} = \omega_p^{2} + c^{2}k^{2}. Reflection below \omega_p explains ionospheric radio bounce.
  • MHD: plasma as conducting fluid. Flux freezing: B lines move with plasma in ideal MHD.
  • Alfveˊnwaves:transverseoscillationsalongBatvA=B/(μ0ρAlfvén waves: transverse oscillations along B at v_{A} = B/\sqrt(\mu_{0}\rho.
  • Fusion:LawsoncriterionnτET>3×1021keV\cdotm−3⋯ .TokamaksapproachthiswithQ≥1Fusion: Lawson criterion n\tau_E T > 3\times10^{21} keV\cdotm^{-3}\cdots. Tokamaks approach this with Q \ge 1.